From (REF ),
\(\partial _{t}\nabla _{ij}h = \nabla _{ij}(\partial _{t}h) = \eta (t)(\sigma _{k}\nabla _{ij}N + \nabla _{j}\sigma _{k}\nabla _{i}N+ \nabla _{i}\sigma _{k}\nabla _{j}N) + N\eta (t)\nabla _{ij}\sigma _{k} - h_{ij},\) 
where
\(\nabla _{ij}\sigma _{k} = \sigma _{k}^{pq,mn}\nabla _{j}b_{pq}\nabla _{i}b_{mn} + \sigma _{k}^{pq}\nabla _{ij}b_{pq}.\) 
By Gauss equation,
\(\nabla _{ij}b_{pq} = \nabla _{pq}b_{ij} + \delta _{ij}\nabla _{pq}h - \delta _{pq}\nabla _{ij}h + \delta _{iq}\nabla _{pj}h - \delta _{pj}\nabla _{iq}h .\) 
Hence
\(\begin{split}\partial _{t}h_{ij} & = N\eta (t)\sigma _{k}^{pq}\nabla _{pq}b_{ij} + k N\eta (t)\sigma _{k}\delta _{ij} - N\eta (t)\sigma _{k}^{pq}\delta _{pq}b_{ij} + N\eta (t)(\sigma _{k}^{ip}b_{jp} - \sigma _{k}^{jp}b_{ip})\\& + N\eta (t)\sigma _{k}^{pq,mn}\nabla _{j}b_{pq}\nabla _{i}b_{mn} + \eta (t)(\sigma _{k}\nabla _{ij}N + \nabla _{j}\sigma _{k}\nabla _{i}N+ \nabla _{i}\sigma _{k}\nabla _{j}N) - h_{ij}.\end{split}\) 
This together with \((\ref {radii})\)  gives the evolution equation of \(b_{ij}\) .
The evolution equation of \(b^{ij} \)  then follows from
\(\partial _{t}b^{ij} = -b^{im}b^{lj}\partial _{t}b_{ml}.\) 
For more details of computations about the evolution equations of \(b_{ij}\)  and \(b^{ij}\) , one can refer to [1]}, [2]}.
Recalling that \( \rho ^{2} = h^{2} + |\nabla h|^{2}\) , we have
\(\begin{split}& \partial _{t}\left(\frac{\rho ^{2}}{2}\right) - N\eta (t)\sigma _{k}^{ij}\nabla _{ij}\left(\frac{\rho ^{2}}{2}\right)\\& = h\partial _{t}h + \nabla _{i}h \nabla _{i}\partial _{t}h - N\eta (t)\sigma _{k}^{ij}\left(h\nabla _{ij}h + \nabla _{i}h\nabla _{j}h + \nabla _{m}h\nabla _{j}\nabla _{mi}h + \nabla _{mi}h\nabla _{mj}h \right)\\& = h\partial _{t}h + \nabla _{i}h\nabla _{i}(N\eta (t)\sigma _{k} - h) \\&- N\eta (t)\sigma _{k}^{ij}\left[\nabla _{i}h\nabla _{j}h + \nabla _{m}h\nabla _{j}(b_{mi} - h\delta _{mi}) \right] - N\eta (t)\sigma _{k}^{ij}h(b_{ij} - h\delta _{ij}) \\& - N\eta (t)\sigma _{k}^{ij}(b_{mi} - h\delta _{mi})(b_{mj} - h\delta _{mj})\\& = (k+1)hN\eta (t)\sigma _{k} - \rho ^{2} + \eta (t)\sigma _{k}\nabla _{i}h\nabla _{i}N - N\eta (t)\sigma _{k}^{ij}b_{mi}b_{mj}.\end{split}\) 
